Virasoro representations with central charges 1/2 and 1 on the real neutral fermion Fock space F^1/2

We study a family of fermionic oscillator representations of the Virasoro algebra via 2-point-local Virasoro fields on the Fock space F^1/2 of a neutral (real) fermion. We obtain the decomposition of F^1/2 as a direct sum of irreducible highest weight Virasoro modules with central charge c = 1. As a corollary we obtain the decomposition of the irreducible highest weight Virasoro modules with central charge c = 1/2 into irreducible highest weight Virasoro modules with central charge c = 1. As an application we show how positive sum (fermionic) character formulas for the Virasoro modules of charge c = 1/2 follow from these decompositions